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hw12 - 6.003 Homework 12 Due at the beginning of recitation...

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6.003 Homework 12 Due at the beginning of recitation on Wednesday, December 2, 2009 . Problems 1. Sampling CT sinusoids Consider 3 CT signals: x 1 ( t ) = cos(3000 t ) , x 2 ( t ) = cos(4000 t ) , and x 3 ( t ) = cos(5000 t ) . Each of these is sampled as follows x 1 [ n ] = x 1 ( nT ) , x 2 [ n ] = x 2 ( nT ) , and x 3 [ n ] = x 3 ( nT ) , where T = 0 . 001 . Which of the resulting DT signals has the highest DT frequency? Which has the lowest DT frequency? 2. Sampling with alternating impulses A CT signal x c ( t ) is converted to a DT signal x d [ n ] as follows: x d [ n ] = x c ( nT ) n even x c ( nT ) n odd a. Assume that the Fourier transform of x c ( t ) is X c ( ) shown below. ω X c ( ) W 1 Determine the DT Fourier transform X d ( e j ) of x d [ n ] . b. Assume that x c ( t ) is bandlimited to W ω W . Determine the maximum value of W for which the original signal x c ( t ) can be reconstructed from the samples x d [ n ] . c. Make a diagram of a system to reconstruct x c ( t ) from x d [ n ] .
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6.003 Homework 12 / Fall 2009 2 3. Boxcar sampling A digital camera focuses light from the environment onto an imaging chip that converts the incident image into a discrete representation composed of pixels. Each pixel represents
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